The most efficient way to pack equally sized spheres isotropically in three dimensions is known as the random close packed state, which provides a starting point for many approximations in physics and engineering. However, the particle size distribution of a real granular material is never monodisperse. Here we present a simple but accurate approximation for the random close packing density of hard spheres of any size distribution based upon a mapping onto a one-dimensional problem. To test this theory we performed extensive simulations for mixtures of elastic spheres with hydrodynamic friction. The simulations show a general (but weak) dependence of the final (essentially hard sphere) packing density on fluid viscosity and on particle size but this can be eliminated by choosing a specific relation between mass and particle size, making the random close packed volume fraction well defined. Our theory agrees well with the simulations for bidisperse, tridisperse, and log-normal distributions and correctly reproduces the exact limits for large size ratios.